Question

how to solve standard form equations? Examples

Answer 1

Standard form equations typically refer to different types of mathematical equations depending on the context. Here, I'll explain how to handle standard form equations in two different contexts: linear equations and quadratic equations.

### 1. Standard Form of Linear Equations

The standard form for a linear equation in two variables is written as:
\[ Ax + By = C \]
where \( A \), \( B \), and \( C \) are integers, and \( A \) and \( B \) are not both zero.

#### Example 1:
Solve the equation \( 2x + 3y = 6 \).

**Steps:**
1. Choose a value for \( x \) (let's say \( x = 0 \)) and solve for \( y \):
\[
2(0) + 3y = 6 \implies 3y = 6 \implies y = 2
\]
So one point is \( (0, 2) \).

2. Choose another value for \( x \) (let's say \( x = 3 \)):
\[
2(3) + 3y = 6 \implies 6 + 3y = 6 \implies 3y = 0 \implies y = 0
\]
So another point is \( (3, 0) \).

3. Plot the points \( (0, 2) \) and \( (3, 0) \) on a graph and draw a line through them to represent the solution set of the equation.

### 2. Standard Form of Quadratic Equations

The standard form for a quadratic equation is written as:
\[ ax^2 + bx + c = 0 \]
where \( a \neq 0 \).

#### Example 2:
Solve the equation \( x^2 - 5x + 6 = 0 \).

**Steps:**
1. Factor the quadratic:
\[
(x - 2)(x - 3) = 0
\]

2. Set each factor to zero:
\[
x - 2 = 0 \implies x = 2
\]
\[
x - 3 = 0 \implies x = 3
\]

3. The solutions to the equation are \( x = 2 \) and \( x = 3 \).

### Additional Methods for Quadratic Equations

You can also solve quadratic equations using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

For the equation \( x^2 - 5x + 6 = 0 \):
- Here, \( a = 1 \), \( b = -5 \), and \( c = 6 \).

1. Calculate the discriminant:
\[
b^2 - 4ac = (-5)^2 - 4(1)(6) = 25 - 24 = 1
\]
2. Now use the quadratic formula:
\[
x = \frac{-(-5) \pm \sqrt{1}}{2(1)} = \frac{5 \pm 1}{2}
\]
3. This gives:
\[
x = \frac{6}{2} = 3 \quad \text{and} \quad x = \frac{4}{2} = 2
\]

### Summary

- For **linear equations**, you can substitute and graph points or manipulate to find the equation of the line.
- For **quadratic equations**, you can factor, complete the square, or use the quadratic formula to find solutions.

If you have a specific standard form equation you'd like to solve or need further clarification on, please let me know!